Math Unit 1

Chapter 1A: Numbers

Chapter 1A: Operations with Negative Numbers

Introduction to Integers

• Definition of Integers: Integers represent whole numbers that can be positive, negative, or zero. They are part of the set of numbers that do not include fractions or decimals.

• Visual Representation:

  • Number Line Example:

  • A number line with integers should be pictured with ticks representing numbers in increasing order such as:

    • … -3, -2, -1, 0, +1, +2, +3 …

• Activity:

  • Add a colored dot on the number line at (-4).

  • Question: How would you get from (-4) to (+2)?

  • This involves moving to the right along the number line.

  • Brainstorming Exercise:

  • Think of as many real-life examples of integers as possible.

  • Examples may include:

    • Temperatures below zero (like -5°C),

    • Bank account balances (like -$20),

    • Elevation levels (like -50 meters), etc.

• Revisiting the Number Line:

  • Represent the jump from (-4) to (+2) in an equation:

  • This can be expressed as: $-4 + 6 = +2$.

  • Reverse the Movement: What if starting at (+2) and ending at (-4)?

  • The equation would be: $2 - 6 = -4$.

Addition and Subtraction

Fundamental Principles

• Adding Negative Numbers:

  • Adding a negative number is equivalent to subtracting its opposite.

• Subtracting Negative Numbers:

  • Subtracting a negative number is equivalent to adding its opposite.

Example Problem

• Example Equation:

  • Calculate (-2) - (-3).

  • Solution Process:

  • By changing subtraction of a negative to addition, it becomes: $-2 + 3 = 1$.

Practice Problems

1. Calculate each expression:

   • a) $4 + (-9)$

   • b) $4 - (-9)$

   • c) $-3 + (-5)$

   • d) $(-3) - (-5)$

Multiplication and Division

Understanding Multiplication

• Concept of Multiplication:

  • Multiplication can be considered as repeated addition.

  • Example Demonstration:

  • The equation $3 imes 4$ can be understood as 3 groups of 4, equaling 12:

    • $3 imes 4 = 12$.

Multiplication Table

• Fill in the table based on rules presented:

  • 3 x (-4) = ???

Multiplication Rules

• General Rules for Multiplication:

  • (positive) x (positive) = (positive)

  • (positive) x (negative) = (negative)

  • (negative) x (positive) = (negative)

  • (negative) x (negative) = (positive)

Division Rules

• General Rules for Division:

  • (positive) ÷ (positive) = (positive)

  • (positive) ÷ (negative) = (negative)

  • (negative) ÷ (positive) = (negative)

  • (negative) ÷ (negative) = (positive)

Practice Problems

2. Solve the following:

   • e) $4\cdot(-9)$

   • f) $-\frac{81}{-9}$

   • g) $-3 imes -5$

   • h) $(-32) ÷ 8$

Chapter 1B: Exponents

Definition of Exponents

• Definition: Exponents are a way to express a number (the base) raised to the power of another number (the exponent), indicating how many times to multiply the base by itself.

Example of Exponents

• Express as Exponent:

  • Rewrite the expression $3 imes 3 imes 3 imes 3$ using exponents, which equals $3^4$.

Practice Exercises

• Rewrite the following expressions using expanded notation and then compute:

  • a) $4^3$

  • b) $(-6)^2$

  • c) $(-2)^5$

  • d) $2^3 imes 3^2$

Summary of Exponents

• Expanded Notation Interpretation:

  • Each expression involves taking the base and multiplying it by itself according to the exponent provided, showcasing foundational understanding in exponential operations.

Chapter IC: Factors

• Definition of Factors:

  • Factors are the positive integers that divide evenly into a given number.

  • Example: For the number 65, the equation $(65 ext{ mod } 5 = 0)$ implies that 5 is a factor since the division $(65 ext{ divided by } 5 = 13)$ yields an integer.

  • Therefore, 5 and 13 are the factors of 65.

Chapter 1D: Numbers

• Prime Factorization:

  • When a number is expressed as a product of its factors, it is referred to as being factorized.

  • The two numbers that multiply to create the larger number are known as a factor pair.

  • Example of a factorization: 5 x 13 = 65.

• Practice Exercise:

  • a) Write out the factor pairs of 24:

  • 1 x 24

  • 2 x 12

  • 3 x 8

  • 4 x 6

  • b) Explain why 8 is not a factor of 30:

  • The operation of $(30 ext{ divided by } 8)$ does not yield a whole number, since $(30 ext{ divided by } 8 = 3.75)$. Because 3.75 is not an integer, 8 is not a factor of 30.

PRIME AND COMPOSITE NUMBERS

• Definition of Prime Numbers:

  • A positive integer is classified as a prime number if it has exactly two factors: 1 and itself.

• Definition of Composite Numbers:

  • A positive integer is classified as a composite number if it has more than two factors.

• Prime Factorization Process:

  • The process of listing out the product of the prime factors of a number is known as prime factorization.

  • Example: The number 12 can be expressed as $(2 imes 2 imes 3) = 12$ or in a different way as $(2^2 imes 3) = 12$.

Chapter 1E: Highest Common Factor

HIGHEST COMMON FACTOR (HCF)

• Definition of HCF:

  • The Highest Common Factor (HCF) also known as the Greatest Common Divisor (GCD) of two or more numbers is defined as the largest factor that is common to all the numbers involved.

• Example Calculation of HCF:

  • To find the HCF of 18 and 45, we determine the factors for each:

  • Factors of 18: 1, 2, 3, 6, 9, 18

  • Factors of 45: 1, 3, 5, 9, 15, 45

  • The largest factor common to both sets is 9, therefore, the HCF of 18 and 45 is 9.

FACTORIZATION TECHNIQUES

• Utilizing Factorization for Larger Numbers:

  • For larger numbers, prime factorization techniques can be employed to find the HCF.

  • Example: To find the HCF of 180 and 324:

  • First, find the prime factorization of each number:

    • For 180:

    • $(180=2\cdot90)$

    • $(90=2\cdot45)$

    • $(45=3\cdot15)$

    • $(15=3\cdot5)$

    • Hence, $(180=2^2\cdot3^2\cdot5)$ .

    • For 324:

    • $(324=2\cdot162)$

    • $(162=2\cdot81)$

    • $(81=3\cdot27)$

    • $(27=3\cdot9)$

    • $(9=3\cdot3)$

    • Therefore, $(324=2^2\cdot3^4)$ .

  • After obtaining prime factorizations, one can further calculate the HCF by taking the lowest power of each common prime factor.

Chapter IF: Multiples

• Definition of Multiples: The multiples of a number are found by counting by that number.

• Lowest Common Multiple (LCM): The smallest multiple that is common to two or more numbers.

Method #1: Finding LCM of 6 and 8

• Calculate the multiples of both numbers and identify the smallest common one.

Chapter IG: Order of Operations

• Key Rule: When an expression contains brackets within brackets, solve the innermost set first.

Practice Problems:

• a) Evaluate: 4 + (3 × (2 + 4))

• b) Simplify: 3² - 15 ÷ 5 + 5

• c) Compute: 5 + -8 × 3

• d) Work through: 15 - (18 - 3) + 6²

Homework Assignments

• For Chapter IF (pg. 21):

  • Problems: #3, 4ace, 5ace, 7, Challenge #9

• For Chapter IG (pg. 23):

  • Problems: #lacegik, 2acegi, 4ace, 6ace, 8ace

Chapter IH: Problem Solving

• Focus: Understanding what operations to perform and in what order.

• Use information to write a mathematical expression.

Example Problem:

• Scenario: For her birthday, Moyan received five $10 notes, three $20 notes, and one $50 note.

  • a) Write an expression for the total amount of money Moyan received.

  • b) Calculate the total amount she received.

Example 13: Sharing Marbles

• Scenario: Gemma bought 2 bags of 50 marbles (total 100), and Jerome bought 3 bags of 35 marbles (total 105). They share the marbles equally among themselves and 3 other friends (4 children total).

  • a) Write an expression for the number of marbles each child received.

  • b) Calculate the actual number of marbles each child received.

Chapter 3A: Fractions

Fractions Overview

• Introduction to Fractions

  • Fractions consist of two parts:

  • Numerator: The top part of a fraction.

  • Denominator: The bottom part of a fraction.

Types of Fractions

• Proper Fractions:

  • Definition: Fractions where the numerator is less than the denominator.

  • Example:

  • $\frac{3}{25}$

• Improper Fractions:

  • Definition: Fractions where the numerator is greater than or equal to the denominator.

  • Example:

  • $\frac{3}{2}$

• Mixed Numbers:

  • Definition: A whole number combined with a proper fraction.

  • Example:

  • $\frac{3}{20}$ can be represented as

  • $3$ (whole number)

  • $\frac{5}{25}$ (proper fraction)

Chapter 3B: Equal Fractions

Equal Fractions

• Two fractions are equal if they have the same value

• Multiplying or dividing both the numerator and denominator by the same non zero number produces an equal fraction

• A fraction in lowest/simplest terms is written with the smallest possible integer numerator and denominator 

• In order to write a fraction in its simplest terms, the numerator and denominator must be divided by their GCF

Practice Problems

• Convert Mixed Numbers and Improper Fractions:

  • Problem 1: Write $3$ as an improper fraction.

  • Solution: $\frac{3}{1}$ or any improper equivalent.

  • Problem 2: Write $\frac{3}{5}$ as a mixed number.

  • Solution: Since $\frac{3}{5}$ is already a proper fraction, it remains the same.

BEDMAS Equation

• Bringing Together REDMAS and Fractions:

  • BEDMAS (or PEMDAS) is a mnemonic for the order of operations:

  • B: Brackets (B for brackets)

  • E: Exponents

  • D: Division

  • M: Multiplication

  • A: Addition

  • S: Subtraction

  • In relation to fractions, understanding that you can multiply or divide both the numerator and denominator by the same number helps in simplifying fractions.

Simplifying Fractions

• Definition of Simplifying:

  • To write a fraction in its simplest or lowest terms.

• Process:

  • A fraction is written in simplest form when both the numerator and denominator are the smallest possible integers.

  • To achieve this, divide both the numerator and the denominator by their Greatest Common Factor (GCF).

• Example:

  • To simplify $\frac{4}{32}$, you would find the GCF of 4 and 32, which is 4.

  • $\frac{4 \div 4}{32 \div 4} = \frac{1}{8}$

Chapter 3C: Adding and Subtracting Fractions

Adding and Subtracting Fractions

Basic Principles

• Condition for Addition/Subtraction:

  • In order to add or subtract fractions, they must have the same denominator.

Steps to Add/Subtract Fractions:

1. Identify the denominators of the fractions to be added or subtracted.

2. Find the Lowest Common Multiple (LCM) of the two denominators.

3. Create equal fractions by adjusting the numerators accordingly, ensuring both fractions now have the LCM as their denominator.

4. After the fractions have been adjusted, you can proceed to add or subtract the fractions as required.

5. Finally, simplify the resulting fraction, if possible.

Example Practice Problems

• Solve the following:
  a) $\frac{3}{2} + \frac{3}{4}$
  b) $\frac{7}{3} - \frac{3}{2}$
  c) $\frac{4}{3} + \frac{3}{4}$
  d) $\frac{8}{3} - \frac{12}{2}$

Solutions

• Example calculations yield the following (not fully worked out for practice):

  • $\frac{21}{12}$ for one problem.

  • For another, $\frac{24 + 24 = 27 - 12 = 1}{8}$

Chapter 3D: Multiplying Fractions

Multiplying Fractions

Basic Principles

• Multiplication of Two Fractions:

  • To multiply two fractions, multiply the numerators together.

  • Multiply the denominators together.

  • Simplify if possible.

Example Calculation

• The mathematical expression for multiplying fractions would appear as follows:

  • Example: $\frac{3}{2} \times \frac{4}{3} = \frac{3 \times 4}{2 \times 3} = \frac{12}{6} = 2$

Multiplying a Fraction by a Whole Number

• Process:

  • When multiplying a fraction by a whole number, convert the whole number to a fraction by expressing it with a denominator of 1.

  • Multiply the numerator of the fraction by the whole number.

Example in a Real-World Scenario

• During a season, Joshua scored 2 of his team's 40 goals:

  • Calculation: $\frac{2}{40} = \frac{2 \times 2}{40 \times 2} = \frac{4}{80}$

  • This illustrates the method of multiplying and evaluating the fraction.

Chapter 3E: Dividing Fractions

Dividing Fractions

Basic Principles

• Division of Fractions:

  • To divide by a fraction, you multiply by its reciprocal.

  • The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.

Example Calculation

• Example explaining the division process includes evaluation of the expression:

  • For instance, $1 + 1 = 1 - 1$ can be related to evaluating fractions. Multiply by the reciprocal:

  • Practically, if performing: $\frac{4}{1} ÷ \frac{1}{2}$, this can be transformed into: $\frac{4}{1} \times \frac{2}{1} = 8$.

Practice Problems

• Solve the following:
  a) $4 ÷ 1$
  b) $1 + ÷ \frac{1}{2}$

Results of Sample Calculations

• Computation details yield values such as:

  • For $\frac{4 x 3}{1} = 12$

  • Additionally, scenarios involving fractional parts like $\frac{3}{4} ÷ \frac{1}{4}$ should be evaluated as well.

Chapter 3F: Decimal Numbers

• Decimal Form: The representation of a number that includes a decimal point.

  • Example: 14.062

• Expanded Form: A method of breaking down a number to show the value of each digit.

  • For 14.062:
    $14 + rac{0}{10} + rac{6}{100} + rac{2}{1000}$

• Mixed Number: A whole number combined with a fraction.

  • For 14.062, it can also be understood as 14 - (value of decimal part).

• Place Value Table: Understanding decimal place values can help in effective calculations.

  • Tens: 14

  • Units: 0

  • Tenths: 0

  • Hundredths: 6

  • Thousandths: 2

• Converting Decimals to Fractions:

  • When converting a decimal to a fraction, the answer should always be provided in lowest terms.

  • Example: To write 0.6 as a fraction,
    $0.6 = rac{6}{10} = rac{3}{5}$

• Ordering Decimals:

  • When asked to order decimals from least to greatest, keep in mind that the further you get from the decimal point, the smaller the value.

  • Example:
    4.005 < 4.050 < 4.500

Chapter 3G: Rounding Decimal Numbers

• Rounding Rules: The same rules used when rounding whole numbers apply to decimal numbers as well.

• To round to a particular place value, you must look at the digit immediately to the right of the desired place.

  • If this digit is 0, 1, 2, 3, or 4, round down; keep your digit the same.

  • Example: To round 6.041 to the nearest hundredth:

    • Considering the digit: 4 (which is 0, 1, 2, 3, or 4)

    • Result: 6.04

  • If the digit is 5, 6, 7, 8, or 9, round up and change your digit to the next value up.

  • Example: To round 10.05 to the nearest tenth:

    • Considering the digit: 5

    • Result: 10.1

Chapter 3H: Adding and Subtracting Decimals

• Procedure: When adding or subtracting decimal numbers, it is critical to align the decimal points and corresponding place values.

• Example:

  • To calculate: $15.3 + 9.26$:

  • Align decimals:

    • $15.30$

    • $+0.26$

  • Result: 24.56

• Practice Problems:

  • a) Find $17.05 + 0.063$

  • Align Numbers:

  • $17.05$

  • $+0.063$

  • b) Find $8.014 - 1.1313$

  • Align Numbers:

  • $8.014$

  • $-1.1313$

  • Expected calculation result would be noted herein.

Chapter 3F: Decimal Numbers

• Decimal Form: 14.062

• Expanded Form:

  • $14 + rac{0}{10} + rac{6}{100} + rac{2}{1000}$

• Mixed Number: 14

  • Components:

  • Tens: 1

  • Units: 4

  • Tenths: 0.0

  • Hundredths: 6

  • Thousandths: 2

• Place Value Table:

  • 14.062

• Converting Decimal to Fraction:

  • When converting a decimal to a fraction, the answer must be in lowest terms.

  • Example: Write 0.6 as a fraction:

  • $0.6 = rac{6}{10} = rac{3}{5}$

• Ordering Decimals:

  • When asked to order decimals from least to greatest, remember:

  • The further away a number is from the decimal point, the smaller the value.

  • Example:

  • 4.005 < 4.050 < 4.500

Chapter 3G: Rounding Decimal Numbers

• Rounding Procedure:

  • When rounding decimal numbers, the same rules apply as when rounding whole numbers.

  • To round to a particular place value, examine the digit to the immediate right of that place value.

• Rounding Down:

  • If the digit to the right is 0, 1, 2, 3, or 4, round down and retain the original digit.

  • Example: Round 6.041 to the nearest hundredth:

  • Rounded Value: 6.04

• Rounding Up:

  • If the digit to the right is 5, 6, 7, 8, or 9, round up and increase the original digit by one.

  • Example: Round 10.05 to the nearest tenth:

  • Rounded Value: 10.1

Chapter 3H: Adding and Subtracting Decimals

• Critical Alignment:

  • It is essential to line up the decimal points and corresponding place values when adding or subtracting decimal numbers.

• Example of Addition:

  • Find $15.3 + 9.26$:

  • Alignment:

    • $<br>15.30</p></li><li><p>9.26 <br>$

    • Answer: 24.56

• Practice Problems:

  • a) Find $17.05 + 0.063$:

  • Alignment:

    • $<br>17.05</p></li><li><p>0.063 <br>$

  • b) Find $8.014 - 1.1313$:

  • Alignment:

    • $<br>8.014</p></li><li><p>1.1313 <br>$

  • Outcome: 6.3827